The effect of critical coupling constants on superconductivity enhancement

In this study, we propose a phenomenological model to extend McMillan's results on a coupling strength equal to 2. We investigate possible strategies to enhance superconductivity by tuning the phonon frequency, carrier number, or pressure. In particular, we show that the critical coupling constants corresponding to the phonon frequency, carrier number, or pressure determine whether the variation of the critical temperature is positive or negative. These observations explain the contrasting behavior between weak and strong coupling superconductors and are consistent with experimental observations. We also demonstrate the dome observed in the carrier number effect and pressure effect. Additionally, these critical coupling constants systematically separate superconductivity into three regions: weak, intermediate, and strong coupling. We find that the enhancement strategies for weak and strong coupling regions are opposite, but both inevitably bring superconductivity into the intermediate coupling region. Finally, we propose general zigzag methods for intermediate coupling superconductors to further enhance the critical temperature.


and
The derivative dT c /d� is equal to 0 when is equal to 2. Define the critical coupling constant corresponding to the phonon frequency c =2.
Second, to investigate the dependency of and T c on carrier number Z , Eqs. (1) and (2) can be rewritten as a function of the carrier number Z . Here, the characteristic phonon frequency uses the jellium phonon frequency 39 � = Z 2 e 2 n ion /ǫ 0 M , where ǫ 0 is the permittivity. The explicit form of and T c as a function of Z are and The derivative of Eqs. (5) and (6) with respect to Z are and Similarly, the derivative dT c /dZ is equal to 0 when is equal to 5/3, denoted as Z c . Third, to investigate the dependency of T c on pressure P , the compressibility β = −(1/V ) * (dV /dP) can be adopted to relate the pressure P and the volume V . Using n ion = N ion /V , where N ion is the number of ions; Eqs. (5) and (6) can thus be rewritten into a function of V: and The derivative of Eqs. (9) and (10) with respect to the pressure P are: . www.nature.com/scientificreports/ The critical coupling constant corresponding to the pressure is P c =4/3. The schematic diagram based on the result from Eqs. (1) to (12), which demonstrates the effects of tuning , Z , or P on T c is plotted in Fig. 1.

Results and discussion
The origin of the dome. The results of our derivations reveal two common features: (1) dT c /dX ∝ 1 − X c / , and (2) d /dX < 0 , where X represents , Z , or P . Specifically, feature (1) indicates that there is a sign change of derivative d /dX at X c . If is larger than X c , the effect of varying X on superconductivity is positive with T c increasing, which corresponds to strong coupling superconductivity. On the other hand, if is smaller than X c , then the effect of varying X on superconductivity is negative with T c decreasing, which corresponds to weak coupling superconductivity. These observations explain the contrasting behavior between weak and strong coupling superconductors. Feature (2) is the reason for the dome-like delineation observed in many strong coupling superconductors. Specifically, for strong coupling superconductors, the coupling constant is expected to be larger than X c , to ensure the effect of varying X on superconductivity is positive. In addition, since the derivative of is negative, decreases when the parameter X increases. In particular, once becomes smaller than X c , the effect of varying X on superconductivity becomes negative. This transition from positive to negative demonstrates the dome-like delineation and can be observed when tuning the carrier number or varying the pressure in the experiments.
Critical temperature T c as a function of phonon frequency . Critical temperature T c is influenced when the phonon frequency is changing. Specifically, the maximum T c appears at c = 2 by varying the phonon frequency Ω, which is consistent with McMillan 3 . More precisely, in the > � c region, the sign of dT c /d� is positive, and T c increases when increasing . This region may correspond to cuprate superconductors. Particularly, LaBaCuO, YBaCuO, BiSrCaCuO, HgBaCaCuO, and TlBaCaCuO demonstrate phonon stiffening effect 8,9 . In Table 1 21 . These values are larger than c indicating that T c can be enhanced by the phonon stiffening effect.
Meanwhile, in the < � c region, the coupling constant is smaller than c . Specifically, the sign of dT c /d� is negative and T c decreases when increases. More precisely, this region may correspond to traditional metallic superconductors. Particularly, VCr, ZrRh, NbMo, MoRe, WRe, PbTl, PbBi, Nb 3 Al, and Nb 3 Ge demonstrate the phonon softening effect 3-6 . In Table 1, the value of phonon softening effect superconductors are listed. For example, when the Debye temperature D of the VCr alloy increases from 370 to 470 K, T c decreases from 3.21 to 0.10 K, and decreases from 0.53 to 0.33. Taking another example, when the Debye temperature D of the ZrRh alloy increases from 192 to 244 K, T c decreases from 5.95 to 3.10 K, and decreases from 0.80 to 0.59. The experiment data show that the coupling constant decreases when the phonon frequency increases, which is consistent with our result d /d� < 0 from Eq. (3). A similar effect also appears in nickel-based superconductors 7 , which validates that the superconductivity is enhanced by giant phonon softening.
Critical temperature T c as a function of carrier number Z. The effect of carrier number on superconductivity through varying the alloy composition, doping concentration, or gating voltage has been widely studied Figure 1. A schematic diagram of the critical temperature T c and the critical coupling constants X c corresponding to the phonon frequency , the carrier number Z , and the pressure P . There is a sign change of derivative d /dX at X c . If is larger than X c , the effect of varying X on superconductivity is positive with T c increasing corresponding to strong coupling superconductivity. Meanwhile, if is smaller than X c , then the effect of varying X on superconductivity is negative with T c decreasing corresponding to the weak coupling superconductivity. www.nature.com/scientificreports/ in material physics. Particularly, the dome-like delineation was observed in metallic materials 17 , cuprates [19][20][21] , iron-based systems 22 , and gating thin film materials 24 . More precisely, the maximum T c appears at Z c = 5/3 by varying the carrier number Z.
For conventional superconductors, we take Nb 3 Al 1−x Ge x as an example 43 , which is showing in Fig. 2a. Specifically, when the Ge component rises from x = 0 to x = 0.29, the electrons per atom ratio (e/a) rises from 4.50 to 4.57 and T c increases from 18 to 21 K. The coupling constant of Nb 3 Al 1 is > 1.8 41 , which is greater than Z c and www.nature.com/scientificreports/ is consistent with the positive carrier effect. Moreover, when the Ge component rises from x = 0.29 to x = 1, the e/a ratio rises from 4.57 to 4.75 and T c decreases from 21 to 7 K. The coupling constant of Nb 3 Ge 1 is < 1 6 which is smaller than Z c and is consistent with the negative carrier effect. For unconventional superconductors, we take the hole-doped experiment of Bi 2 Sr 2 Ca 1 Cu 2 O x as an example 21 , which is showing in Fig. 2b. Specifically, when the hole doping h (holes/Cu) rises from h = 0.11 to h = 0.21, the coupling constant monotonically decreases from = 2.66 to = 0.97. Additionally, the critical temperature first rises from 66 K ( = 2.66 > Z c at h = 0.11) to 88 K ( = 2.15 > Z c at h = 0.16), then drops to 77 K ( = 1.5 < Z c at h = 0.20) and further to 67 K ( = 0.97 < Z c at h = 0.21). This dome-like delineation showed that the superconductivity is increasing in the underdoped region with coupling stronger than Z c , and decreasing in the overdoped region with coupling weaker than Z c .
Critical temperature T c as a function of pressure effect P. The effect of pressure on superconductivity is an important topic in the field of condensed matter physics. In particular, for strong coupling superconductors, the pressure effect is positive and may lead to a dome-like delineation at higher pressure. These characteristics were observed in cuprate superconductors 12,44 , iron-based superconductors 13,14,45 , and hydrogen-rich superconductors 15,16 . In Table 2, the value of positive or dome-like pressure effect superconductors are listed. Specifically, the coupling constants of cuprates are greater than P c as mentioned in the previous section. Here, we note the couple constant of iron-based superconductors; for instance, LaFeAsO 1−x F x and Ba(Fe 1−x Co x ) 2 As 2 are = 2.38 and 2.83, respectively 46 . These strong coupling superconductors have coupling strength larger than P c , thus the positive or dome-like delineation of the pressure effect can be observed.
Meanwhile, most metallic superconductors are weak-coupling superconductors. Particularly, Al, Cd, Sn, In, and Pb are examples with negative pressure effect 10,11 . Specifically, the coupling constant of Al, Cd, Sn, In, and Pb are less than P c (see Table 2). Clearly, these superconductors have less than P c . Additionally, the negative pressure effect is also observed in the covalent compound MgB 2 11 , with equal to 0.7 49 , which is smaller than P c .
The strategies of T c enhancement. According to these three critical coupling constants: c , Z c , and P c , superconductors can be classified by their coupling strength: weak coupling ( < P c ), intermediate coupling ( P c < < � c ), and strong coupling ( > � c ). A schematic diagram can be seen in Fig. 3. In the weak coupling region ( < P c ), the coupling constant is less than P c . Particularly, all derivatives of critical temperature dT c /d� , dT c /dZ , and dT c /dP are negative, as shown in Fig. 3. Hence, by decreasing , Z or P , the superconductivity can be enhanced. These procedures increase the coupling constant , because the derivatives of the coupling constant d /d� , d /dZ , and d /dP are negative. We can accurately infer that the superconductivity of weak coupling superconductors is enhanced by increasing the coupling strength. In addition, once the coupling constant becomes greater than P c , the superconductivity enters into the intermediate region, and the tendencies of T c become complicated.
Meanwhile, in the strong coupling region ( > � c ), the coupling constant is larger than c . Particularly, all derivatives of the critical temperature dT c /d� , dT c /dZ , and dT c /dP are positive in Fig. 3, thus the superconductivity can be increased by increasing , Z or P . These procedures decrease the coupling constant , such that we can adequately infer that the superconductivity of strong coupling superconductors is enhanced by decreasing the coupling strength. Once the coupling constant is smaller than c , the superconductivity enters into the intermediate coupling region and the tendencies of T c become complicated. Besides, comparing the strong coupling region with the weak coupling region, three tendencies of T c are contrasting between the two regions. Table 2. Examples of pressure effect P on superconductivity and value. The critical coupling constant corresponding to the pressure is P c = 4/3. The positive or dome-like pressure effect appears when > P c , and the negative pressure effect appears when < P c .

Compound
Positive or dome-like , the enhancement methods are more interesting. In particular, take that belongs in the interval P c < < Z c for the following discussion without loss of generality. In this interval, superconductivity can be enhanced by increasing P or decreasing Z . First, let the superconductivity be optimized by tuning Z , such that is equal to Z c , denoted as 1 . Second, since 1 is larger than P c , we can increase P to increase T c . The second step causes to decrease and T c is optimized when = P c , denoted as 2 . Third, now 2 is smaller than Z c , T c can be further increased by decreasing Z . The third step increases and T c is optimized when = Z c , denoted as 3 . Repeat step 2 and step 3 by increasing P and decreasing Z alternately; T c can be enhanced like a zigzag mountain climbing. In this study, we propose simultaneously gating and pressurizing on thin-film superconductors to verify our discussion. Furthermore, FeSe has been observed under gating 25 and pressurizing 50 independently. More precisely, the negative carrier number effect and the dome-like pressure effect suggest the coupling strength P c < FeSe < Z c , which agree with FeSe = 1.6 47 . Additionally, we propose that increasing P and decreasing alternately, or increasing Z and decreasing alternately, are two other methods to enhance superconductivity in the intermediate region.

Conclusion
In this study, we proposed a phenomenological model based on phonon-mediated interaction, which explains the difference between weak and strong coupling superconductors affected by tuning phonon frequency , carrier number Z , and pressure P . We introduced the concept of critical coupling constants and enhancement strategies for superconductivity, extending McMillan's results on coupling strength equal to 2. Specifically, the sign of the first-order derivative dT c /dX with respect to X = , Z , or P , indicates that T c is increasing or decreasing when either these three parameters change. More precisely, these three derivatives have two features in common: (1) the coupling constant beyond (or below) the critical coupling constant X c determined dT c /dX to be positive (or negative), and (2) the dome-like delineation observed in strong coupling superconductors because d /dX is always negative. Overall, these observations explain the differences between weak and strong superconductors.
Furthermore, using three critical coupling constants c , Z c , and P c , superconductors can be classified by their coupling strength and consequently, correspond to different enhancement strategies. Specifically, for the weak coupling region ( < P c ), T c can be increased by decreasing , Z , and P , causing to be increased, resulting in intermediate coupling. In contrast, superconductors in the strong coupling region ( > � c ) can be enhanced by increasing , Z , and P , causing to be decreased, resulting in intermediate coupling. Moreover, for superconductors in the intermediate coupling region ( P c < < � c ), the zigzag strategies may further enhance superconductivity.

Data availability
The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.